Abstract
Advances in heterostructure fabrication have opened new frontiers in moiré physics. Here we extend moiré engineering from artificially assembled thin flakes with mismatched lattice parameters to materials that host incommensurate orders, presenting a long-period moiré superlattice in a layered charge-density-wave compound, EuTe4. Using high-momentum-resolution X-ray diffraction, we found two coexisting incommensurate charge density waves with slightly mismatched in-plane wavevectors. The interaction between these two charge density waves leads to joint commensuration with the lattice and a moiré superstructure with a period of ~13.6 nm, offering key insights into the unique properties of EuTe4, such as the temperature-invariant incommensurate wavevectors and unconventional in-gap states. Owing to interlayer phase shifts, the moiré superstructure exhibits a clear thermal hysteresis, accounting for the large hysteresis in electrical resistivity and numerous metastable states. Our findings open new directions for moiré engineering based on incommensurate lattices and highlight the important role of interlayer ordering in stacked structures.
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Data availability
The experimental data associated with this paper are available via the Harvard Dataverse at https://doi.org/10.7910/DVN/BVWZVL (ref. 49). Source data are provided with this paper.
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Acknowledgements
We thank R. Comin, A. Kogar, H. Ning, K. H. Oh and D. Shi for helpful discussions. The work at MIT was supported by the US Department of Energy, the BES DMSE (data collection and analysis) and the Gordon and Betty Moore Foundation’s EPiQS Initiative grant GBMF9459 (manuscript writing). Research conducted at the Center for High-Energy X-ray Science (CHEXS) is supported by the National Science Foundation (BIO, ENG and MPS Directorates) under award number DMR-2342336. B.L. acknowledges support from the Ministry of Science and Technology of China (grant number 2023YFA1407400), the National Natural Science Foundation of China (grant number 12374063) and the Shanghai Natural Science Fund for Original Exploration Program (grant number 23ZR1479900). N.L.W. acknowledges support from the National Natural Science Foundation of China (grant number 12488201), and the National Key Research and Development Program of China (grant numbers 2024YFA1408700 and 2022YFA1403901). D.W. acknowledges support from the National Key Research and Development Program of China (grant number 2024YFA1408700). N.F.Q.Y. acknowledges support from the National Natural Science Foundation of China (grant number 12174021).
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B.Q.L., Y.S. and A.Z. conceived the study. B.Q.L., Y.S., A.Z., S.S. and J.P.C.R. performed the XRD measurements. Q.L., D.W. and N.L.W. synthesized, characterized and prepared the EuTe4 crystals. S.S. and J.P.C.R. maintained and set up the synchrotron end-station at CHESS. N.F.Q.Y developed the mean-field theory. B.Q.L., Y.S. and A.Z. analysed the data with the help of J.L., Z.N. and S.M. B.Q.L., Y.S., A.Z. and N.F.Q.Y. wrote the paper with critical input from N.G. and all other authors. The work was supervised by N.G.
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Extended data
Extended Data Fig. 1 Diffraction data acquire with bulk crystal sample.
(2 KL) cut of the reciprocal space mapping data acquired in the same experimental geometry. the result is, in general, consistent with the results shown in Fig. 2a. The CDW satellite peaks at (2 q1 L) (L = integer) and (2 q2 L) (L = half integer) can be produced. The δ and δ’ peaks can also be clearly resolved near the Bragg peaks. We note that the δ peaks have a broader peak profile along the L axis compared to δ peaks, which is also consistent with the results demonstrated in the main text. The only difference between the bulk results and the flake results is the peak elongation along the L direction in bulk results. This is due to the stacking faults in bulk crystals. Thus, the bulk measurement is essentially equivalent to measuring a stack of EuTe4 flakes with random relative phases across different flakes. This would cause diffraction intensity to emerge in non-integer (and non-half-integer) L.
Extended Data Fig. 2 Optical image of the measured sample.
Optical micrograph of the EuTe4 flake measured in the X-ray diffraction experiment. The long and straight edge of the flake is parallel to the a-axis of the crystal. A clear and straight edge usually indicates good crystallinity for EuTe4. Successful exfoliation of single-domain EuTe4 flakes is pivotal to the success of the experiment. We note that amorphous glass cover slides are chosen as the substrates in order to avoid extra diffraction peaks in the reciprocal space mapping experiment. The scales bar corresponds to 100 μm.
Extended Data Fig. 3 Classification of possible CDW configurations in EuTe4.
The CDW configurations in EuTe4 can be classified by the respective out-of-plane period or wavevector of monolayer and bilayer CDWs. The phase (0 or π) of CDW in each layer of Te sheet is represented by an arrow (up or down). According to the experimental observation, the ground state features 1c periodicity for monolayer (q1,c = c*) and 2c periodicity for bilayer (q2,c = 0.5 c*). All configurations satisfying the out-of-plane wavevector condition are grouped into ground state G. Each column in the group represents a different configuration differentiated by the relative phase between the monolayer and bilayer. All other ground-state configurations indistinguishable in diffraction measurement are equivalent to one of these two columns in terms the out-of-plane period and relative phases between layers. Similarly, M1, M2, and M3 correspond to metastable states classified by different out-of-plane periodicity respectively.
Extended Data Fig. 4 Simulated diffraction patterns.
Simulated diffraction pattern [(2,K,L) cut] of the ground state (a) CDW configuration as well as the metastable state M1 (b), M2 (c), and M3 (d), as indicated in Extended Data Fig. 3. The simulated diffraction patterns are generated by the Fourier transform amplitude of an electron density distribution model. In this model, only Te square-net sheets, where CDWs are originated, are considered. Each Te atom is modeled as a 3D Gaussian distribution of electrons. Each Te atom is then displaced by CDWs, expressed as \(u=A\cos (2\pi {{\bf{q}}}_{{{i}}}\cdot {\bf{b}})\), where A = 0.4 Å is the CDW amplitude. qi is the CDW wavevector (i = 1, 2), where the value of qi is taken from experimental values. b is the lattice parameter. The simulation considers up to the first order of CDW satellites. The fringes in the vicinity of peaks with large intensity is a finite-size effect in the fast Fourier transform process.
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Lv, B.Q., Su, Y., Zong, A. et al. Large moiré superstructure of stacked incommensurate charge density waves. Nat. Mater. 25, 420–426 (2026). https://doi.org/10.1038/s41563-025-02360-1
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DOI: https://doi.org/10.1038/s41563-025-02360-1
